Problem: A group of adults and kids went to see a movie. Tickets cost $$7.50$ each for adults and $$2.50$ each for kids, and the group paid $$47.50$ in total. There were $3$ fewer adults than kids in the group. Find the number of adults and kids in the group.
Explanation: Let $x$ equal the number of adults and $y$ equal the number of kids. The system of equations is then: ${7.5x+2.5y = 47.5}$ ${x = y-3}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${y-3}$ for $x$ in the first equation. ${7.5}{(y-3)}{+ 2.5y = 47.5}$ Simplify and solve for $y$ $ 7.5y-22.5 + 2.5y = 47.5 $ $ 10y-22.5 = 47.5 $ $ 10y = 70 $ $ y = \dfrac{70}{10} $ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into ${x = y-3}$ to find $x$ ${x = }{(7)}{ - 3}$ ${x = 4}$ You can also plug ${y = 7}$ into ${7.5x+2.5y = 47.5}$ and get the same answer for $x$ ${7.5x + 2.5}{(7)}{= 47.5}$ ${x = 4}$ There were $4$ adults and $7$ kids.